1. **State the problem:** Find the center of mass of the pair (A, B) given the positions and masses of points A and B. 2. **Given data:** - Point A is at the origin $(0,0)$. - Point B is at an angle of $35^\circ$ from the horizontal, distance $L_1 = 18.0$ m from A. - Mass at B: $m_B = 12.0$ kg. - Mass at A: Since not given, assume $m_A$ is the mass at A (to be found). - The center of mass of (A, B) is $12.2$ m from A along the line connecting A and B. 3. **Formula for center of mass along a line:** $$ R_{cm} = \frac{m_A \cdot 0 + m_B \cdot L_1}{m_A + m_B} $$ where $R_{cm}$ is the distance from A to the center of mass. 4. **Substitute known values:** $$ 12.2 = \frac{m_A \cdot 0 + 12.0 \times 18.0}{m_A + 12.0} $$ 5. **Solve for $m_A$:** Multiply both sides by $(m_A + 12.0)$: $$ 12.2(m_A + 12.0) = 12.0 \times 18.0 $$ $$ 12.2 m_A + 146.4 = 216 $$ Subtract 146.4 from both sides: $$ 12.2 m_A = 216 - 146.4 = 69.6 $$ Divide both sides by 12.2: $$ m_A = \frac{69.6}{12.2} $$ $$ m_A = \cancel{\frac{69.6}{12.2}} = 5.7049 \approx 5.7\ \text{kg} $$ 6. **Interpretation:** The mass at point A is approximately 5.7 kg to have the center of mass of the pair (A, B) at 12.2 m from A. **Final answer:** $$m_A \approx 5.7\ \text{kg}$$